Solutions to Exercises
141
that the contribution from scattering states with energies smaller than that of
the resonant state is practically negligible due to the confining effect of the
Coulomb barrier.
(ii) One expands here a 2p 3/2 state (E = 1.905 MeV and Γ =61.89 keV;
V 0 = 70 MeV) in a Woods-Saxon basis containing the bound 2p 3/2 level
(E = −0.0923 MeV; V
(B)
0
= 75 MeV). As a consequence, the resonance
width has to be brought by the scattering states. Nevertheless, the component
of the 2p 3/2 state of the basis is still close to one, whereas the continuum
component plays a secondary role. Once again, one can see an effect of the
Coulomb barrier: even if the expanded 2p 3/2 state is unbound, its wave function
is localized and has a large overlap with the bound 2p 3/2 basis state. The
diagonalization yields an almost exact result as well.
(iii) One now deals with the case of a 2p 3/2 state that is bound in both potentials.
Here V 0 = 80 MeV and V
(B)
0 =75 MeV, and the 2p 3/2 state lies at E= –
0.0923 MeV and E= –2.569 MeV, respectively. The scattering component in
this case is almost negligible, which reflects the localized character of bound
proton states. After the diagonalization, one obtains E=–2.569 MeV and a
negligible value of Γ for the 2p 3/2 state, which is indeed very close to the
exact result.
Exercise X.
A. One notices from calculations that the bound eigenstates of Woods-Saxon
potentials are well reproduced using a basis generated by a Pöschl-TellerGinocchio potential. Nevertheless, one cannot expand unbound states using the
latter basis in practice. This arises because the Pöschl-Teller-Ginocchio potential
bears neither centrifugal nor Coulomb barriers, necessary to generate resonance
states of low energy. Thus, as the basis generated by a Pöschl-Teller-Ginocchio
potential does not have physical asymptotic properties, it cannot provide with
expansions of resonance states.
B. The set of bound and real-energy scattering eigenstates generated by the PöschlTeller-Ginocchio potential is complete in the domain of bound states. Hence,
loosely bound states can also be expanded with this set of states. If one decreases
the potential depth of the Woods-Saxon potential by a small amount, one can still
expand the considered eigenstate of the Woods-Saxon potential with a Berggren
basis generated by a Pöschl-Teller-Ginocchio potential. Therefore, the expansion
of this loosely bound state with a basis of eigenstates generated by Pöschl-TellerGinocchio potential is still precise.
C. The single-particle basis generated by the Pöschl-Teller-Ginocchio contains
bound states and real-energy scattering states. Consequently, the resonance
states cannot be expanded therein. Moreover, even if the width of a resonance
is narrow, a resonance still increases exponentially in modulus on the real axis.
As the completeness relation generated by a Pöschl-Teller-Ginocchio potential
can only expand bound states, it is impossible to diagonalize resonance states
therein.
141
that the contribution from scattering states with energies smaller than that of
the resonant state is practically negligible due to the confining effect of the
Coulomb barrier.
(ii) One expands here a 2p 3/2 state (E = 1.905 MeV and Γ =61.89 keV;
V 0 = 70 MeV) in a Woods-Saxon basis containing the bound 2p 3/2 level
(E = −0.0923 MeV; V
(B)
0
= 75 MeV). As a consequence, the resonance
width has to be brought by the scattering states. Nevertheless, the component
of the 2p 3/2 state of the basis is still close to one, whereas the continuum
component plays a secondary role. Once again, one can see an effect of the
Coulomb barrier: even if the expanded 2p 3/2 state is unbound, its wave function
is localized and has a large overlap with the bound 2p 3/2 basis state. The
diagonalization yields an almost exact result as well.
(iii) One now deals with the case of a 2p 3/2 state that is bound in both potentials.
Here V 0 = 80 MeV and V
(B)
0 =75 MeV, and the 2p 3/2 state lies at E= –
0.0923 MeV and E= –2.569 MeV, respectively. The scattering component in
this case is almost negligible, which reflects the localized character of bound
proton states. After the diagonalization, one obtains E=–2.569 MeV and a
negligible value of Γ for the 2p 3/2 state, which is indeed very close to the
exact result.
Exercise X.
A. One notices from calculations that the bound eigenstates of Woods-Saxon
potentials are well reproduced using a basis generated by a Pöschl-TellerGinocchio potential. Nevertheless, one cannot expand unbound states using the
latter basis in practice. This arises because the Pöschl-Teller-Ginocchio potential
bears neither centrifugal nor Coulomb barriers, necessary to generate resonance
states of low energy. Thus, as the basis generated by a Pöschl-Teller-Ginocchio
potential does not have physical asymptotic properties, it cannot provide with
expansions of resonance states.
B. The set of bound and real-energy scattering eigenstates generated by the PöschlTeller-Ginocchio potential is complete in the domain of bound states. Hence,
loosely bound states can also be expanded with this set of states. If one decreases
the potential depth of the Woods-Saxon potential by a small amount, one can still
expand the considered eigenstate of the Woods-Saxon potential with a Berggren
basis generated by a Pöschl-Teller-Ginocchio potential. Therefore, the expansion
of this loosely bound state with a basis of eigenstates generated by Pöschl-TellerGinocchio potential is still precise.
C. The single-particle basis generated by the Pöschl-Teller-Ginocchio contains
bound states and real-energy scattering states. Consequently, the resonance
states cannot be expanded therein. Moreover, even if the width of a resonance
is narrow, a resonance still increases exponentially in modulus on the real axis.
As the completeness relation generated by a Pöschl-Teller-Ginocchio potential
can only expand bound states, it is impossible to diagonalize resonance states
therein.